I have the week off for Thanksgiving, so some friends and I have spent the last few days in Napa hanging out and doing some wine tasting. We got a recommendation to visit Freemark Abbey Winery, which we did, where I came across what seems like a pretty good problem-based investigation. Here is the scenario as they describe it (based on actual events):

Basically, they had to decide: harvest the grapes now and guarantee the production of a moderately priced wine, OR wait for the incoming storm and hope that it encourages the mold growth necessary for production of high end wine. There are some accompanying details that the winery used in the decision making process (which I have somewhat shortened here):

• There was a 50% chance the storm would hit Napa Valley
• There was a 40% chance that, if the storm did strike, it would lead to the development of botrytis mold
• If Jaeger pulled grapes before the storm, he could produce wine that would sell for $2.85 per bottle
  
• If he didn't pull and the storm DID strike (but didn't produce the mold), he could produce wine that would sell for $2.00 per bottle
• If he didn't pull and the storm DID strike (and did produce the mold), he could produce wine that would sell for $8.00 per bottle (although at 30% less volume because of the process to produce this wine)
  
• If he didn't pull and the storm DIDN'T strike, he could produce wine that would sell for $3.50 per bottle

I can imagine several different versions of the problem depending on the group of students you were working with (by adding/removing details and alternative scenarios). I am attaching the printouts that I photographed in case anyone wants the details.

The 12th graders and I started a new unit/project/problem/whatever last week. It started with this image:

We took some guesses before someone correctly identified the image as the readout from a stationary bike machine. If this was the readout, I nudged, then:

I let them play with it for a while. They identified some important questions they had and information they wanted:

• What do the dots represent?
• How can we quantify this?
• Does speed matter?
• Did the bike have gears?
• What do we mean when we say, "what does the course look like?

Ultimately, they agreed upon some answers and, as a result, gave themselves some space to work within. This part takes time and I really think we need to give it the time it deserves. If we don't let students pose and answer those questions, the rest of the unit suffers. I let them know that we would examine some simpler cases first and return to the unit question somewhere down the road.

So, the next day they came back and and I had them map the course for these two:

This took another class period for them to sort out. By the end, though, they started to feel pretty good about the fact that the dots were a measure of resistance which they connected to the slope of the course at that point. Towards the end of class, someone asked, "What would it look like if you were going downhill?' They talked for a bit and devised a system to handle that, so the next day I gave them this:

This allowed them to have conversations about what it meant when the resistance was positive, what it means when resistance is zero, and what it means when resistance is negative.

It's fun to watch the students construct an understanding of antiderivatives. The three-day span reminded me of a few things:

• the importance of listening to how students are thinking about and making sense of problems

• our role as teachers in responding to that way of knowing by bringing what might be the "next good problem" for them
• allowing them the time to sort things out together

I feel like there are a couple ways to go with this trajectory next:

1. Stay with resistance, but use a continuous curve
2. Stay with this discrete model, but switch contexts (maybe speed or rate of growth)
3. Stay with discrete and resistance, but use a more complicated readout

I'm leaning towards #1. What do you think would be best for the students?

Our Juniors are away on Internship right now so I have been driving all over the city checking in on them and seeing what they are up to. With all that time in the car I was doing some serious project brainstorming for next year. I wanted to put my ideas down in a place where I could come back to them (especially because some of these are still real rough drafts). I'm sticking to the problem-based format because I like it...a LOT. In each of these, students would have to create the mathematics as it emerges in their exploration with the problem.  I would love some blogger critique so please let me know what you think...